Polylogarithm sheaves In many different places, I could find the notion on ''(poly)logarithm sheaves''. As is indicated in the name of it, I guess that it should have something to do with (poly)logarithm function: $\mathrm{Li}_s(z)$. How are they related to each other? Any reference would be helpful, but it would be better if it requires less preliminaries. 
 A: I would suggest looking at Hain's article (arXiv version here)
R. Hain. Classical polylogarithms. Motives Proceedings, vol II, Proc. Symposia Pure Math 55.2, 1994. 
Very roughly, one writes out a multi-valued function on $\mathbb{P}^1\setminus\{0,1,\infty\}$ which takes values in ${\rm GL}_{n+1}(\mathbb{C})$ where the entries of the matrix are built from (poly)logarithms up to ${\rm Li}_n$. This is the fundamental solution of a linear differential equation. The monodromy of the differential equation gives rise to a local system on $\mathbb{P}^1\setminus\{0,1,\infty\}$. This is the $n$-th polylogarithm local system (which can equivalently be regarded as a sheaf). Actually, doing this for all $n$ gives rise to a pro-local system which you might call the polylogarithm sheaf. Under the correspondence between local systems and representations of the fundamental group, this corresponds to the representation of $\pi_1(\mathbb{P}^1\setminus\{0,1,\infty\})$ on the completion of its group ring w.r.t. the augmentation ideal. 
This has a Hodge-theoretic and even motivic refinement. These are relevant when trying to express regulators on K-theory in terms of polylogarithms (on the way to formulas for special L-values). 
You can find all this and more in Hain's article, also lots of references in there. Unfortunately, this is not quite the discussion that would "require less preliminaries"... 
